sooo, if anyone is in engineering or is simply taking a class similar to this, can you explain to me what an axiom is. like i think i understand that they are simply a set of rules for a vector, but how do they prove if something is a vector space? for example, I'm given the matrix |a 4| and asked to find if it is a vector space |4 b|
Help with linear algebra (math 102)
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FreedGucci
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first_rodeo
Active member
why such basic math OP did you go to HS in like West Virginia.
i didnt even do advanced math and didnt have to take anything before business calc
i didnt even do advanced math and didnt have to take anything before business calc
_DUSTY_
Active member
I can't answer your question directly since I honestly haven't done much with linear algebra since I finished the course besides some very basic matrix. try looking up some axiom proofs on google. Like rude boy said, to prove something is a vector space you need to satisfy every axiom. Most of the proofs on the internet deal with vectors however you can utilize matrix addition instead of vector addition and I'm sure you know how to deal with multiplying a matrix by a scalar. Also if I recall correctly you can use other axioms to prove certain ones (eg. you can use associativity to prove inverse elements.) Also there are websites out there you can join such as physicsforums.com with people who will be able to help you more on these topics than ns.
DasGoatenshafter
Member
An axiom is kinda like a theorem. Its a rule of math. So, when proving something, like if something is a vector space, or is linearly independent, you would go through with proofs that show whatever you're working with satisfies the given rules or axioms.
Arabian
Member
Let S be a non-empty set. A set is just a collection of elements with no structure. You can't do anything interesting with them. It isn't until we equip S with a binary operation (addition, multiplication, composition, convolution etc.) defined by *:SxS -> S such that (a, b) maps to a*b under the operation that we can start to say anything meaningful about S. When we have such a rule, we look at the way * operates on the elements of the set. We can ask a few interesting questions:
Is a*b in S for every combination of a and b in S?
Is there some special element (say, e) so that a*e = a = e*a for every element a in S?
Is there a special element a' in S so that a*a' = a'*a = e in S?
What about equipping S with more than one binary operation?
A vector space V is such a structure. A vector space is a set V equipped with multiplication (*) and addition (+) so that we have what amounts to two separate structures on V with respect to each binary operation. The full tuple that defines a vector space V would be (V, +, *, 0, 1, k) where V satisfies:
For (V, +, 0):
1) u + v is in V for every combination of u and v in V.
2) u + v = v + u (which isn't always true for every binary operation. Think matrix multiplication under (*))
3) There exists a u' in V called the "inverse" of u so that u + u' = u' + u = 0.
4) Since 0 is in V, u + 0 = 0 + u = u for every u in V.
And for the structure with respect to (*), we have for (V, *, 1, K) where K is the field over which the vector draws its scalars... (think R. It will almost always be R for your level), we have:
5) u*v is in V for every combination of u, v in V
6) u*(m + n) = u*m + u*n for every u in V, n,m in K. This is just your normal distributivity wr.t. vectors in V. (likewise, (m + n)*u = m*u + n*u. Take care to keep your vectors on the side with which you're multiplying)
7) (u + v)*m = m*(u + v) = m*u + m*v for every u, v in V, m in K.
8. (m*n)*u = m*(n*u) for every u in V, m,n in K.
9. 1*u = u*1 = u for every u in V.
10. 0*u = u*0 = 0 for every u in V.
Now, this may seem confusing, and it certainly is if this is your first time seeing axioms that define a structure, but it makes tons more sense if you suss it out thusly: "When i have a set and an operation on that set, what meaningful things can i do with that binary operation that preserve the structure of that set with respect to that operation?" This is the point of Algebra in general - specifically what you begin to touch upon in the early years of linear algebra. We never stray outside of our set. We always maintain the structure of the set with respect to the binary operations we're given.
Vectors in and of themselves do not exist outside of a vector space, and they're important because it allows us to orient ourselves arbitrarily in space on given basis (you'll learn this eventually). Right now, memorize.
Is a*b in S for every combination of a and b in S?
Is there some special element (say, e) so that a*e = a = e*a for every element a in S?
Is there a special element a' in S so that a*a' = a'*a = e in S?
What about equipping S with more than one binary operation?
A vector space V is such a structure. A vector space is a set V equipped with multiplication (*) and addition (+) so that we have what amounts to two separate structures on V with respect to each binary operation. The full tuple that defines a vector space V would be (V, +, *, 0, 1, k) where V satisfies:
For (V, +, 0):
1) u + v is in V for every combination of u and v in V.
2) u + v = v + u (which isn't always true for every binary operation. Think matrix multiplication under (*))
3) There exists a u' in V called the "inverse" of u so that u + u' = u' + u = 0.
4) Since 0 is in V, u + 0 = 0 + u = u for every u in V.
And for the structure with respect to (*), we have for (V, *, 1, K) where K is the field over which the vector draws its scalars... (think R. It will almost always be R for your level), we have:
5) u*v is in V for every combination of u, v in V
6) u*(m + n) = u*m + u*n for every u in V, n,m in K. This is just your normal distributivity wr.t. vectors in V. (likewise, (m + n)*u = m*u + n*u. Take care to keep your vectors on the side with which you're multiplying)
7) (u + v)*m = m*(u + v) = m*u + m*v for every u, v in V, m in K.
8. (m*n)*u = m*(n*u) for every u in V, m,n in K.
9. 1*u = u*1 = u for every u in V.
10. 0*u = u*0 = 0 for every u in V.
Now, this may seem confusing, and it certainly is if this is your first time seeing axioms that define a structure, but it makes tons more sense if you suss it out thusly: "When i have a set and an operation on that set, what meaningful things can i do with that binary operation that preserve the structure of that set with respect to that operation?" This is the point of Algebra in general - specifically what you begin to touch upon in the early years of linear algebra. We never stray outside of our set. We always maintain the structure of the set with respect to the binary operations we're given.
Vectors in and of themselves do not exist outside of a vector space, and they're important because it allows us to orient ourselves arbitrarily in space on given basis (you'll learn this eventually). Right now, memorize.